Trac #16268: Better normalization for fraction field elements

Normalize the leading coefficient of the denominator to one when
reducing elements of fraction fields of univariate polynomial rings over
fields. Doing so
- often leads to more readable results,
- helps limiting coefficient blow-up during computations with fractions
over complicated base fields (e.g., elements of ℚ(x,y)(t)), typically
leading to much better performance,
- makes hashing more (though not 100%) reliable, see the original
description and the comments below.

The following further improvements are IMO desirable but out of the
scope of this ticket:
- clearing denominators in the numerator and denominator instead of
making the leading coefficient of the denominator monic when that makes
sense (i.e., for printing, and perhaps for computations in nested
rational function fields, but making it fast enough requires some work),
- also normalizing the leading coefficients over non-fields where that
makes sense.

Original description:

If K is a field then K(u), the function field, has a reduce() method
which cancels the gcd but does not put into a canonical form by (for
example) dividing through by the leading coefficient of the denominator
to make the denominator monic.  This means that equal elements may have
different hashes, and hence that putting function field elements into a
set does not work as a mathematician would expect.  For example:
{{{
sage: Ku.<u> = FractionField(PolynomialRing(QQ,'u'))
sage: a = 27*u^2+81*u+243
sage: b = 27*u-81
sage: c = u^2 + 3*u + 9
sage: d = u-3
sage: s = a/b
sage: t = c/d
sage: s==t
True
sage: len(Set([s,t]))
2
}}}

URL: https://trac.sagemath.org/16268
Reported by: cremona
Ticket author(s): Robert Bradshaw, Erik Massop, Marc Mezzarobba
Reviewer(s): Julian Rüth

src/sage/algebras/quatalg/quaternion_algebra_element.pyx

@@ -843,7 +843,7 @@ cdef class QuaternionAlgebraElement_generic(QuaternionAlgebraElement_abstract):
             sage: type(theta)
             <... 'sage.algebras.quatalg.quaternion_algebra_element.QuaternionAlgebraElement_generic'>
             sage: theta._repr_()
-            '1/x + x*i + (-x - 1)*j + (2/(3*x^3 + 5))*k'
+            '1/x + x*i + (-x - 1)*j + (2/3/(x^3 + 5/3))*k'
         """
         return self._do_print(self.x, self.y, self.z, self.w)
 

src/sage/categories/functor.pyx

@@ -262,7 +262,7 @@ cdef class Functor(SageObject):
                             Ring endomorphism of Finite Field in a of size 5^2
                               Defn: a |--> 4*a + 1
             sage: fF((a^2+a)*t^2/(a*t - a^2))
-            3*a*t^2/((4*a + 1)*t + a + 1)
+            ((4*a + 2)*t^2)/(t + a + 4)
 
         """
         try:

src/sage/coding/linear_code.py

@@ -3497,7 +3497,7 @@ def zeta_function(self, name="T"):
 
             sage: C = codes.HammingCode(GF(2), 3)
             sage: C.zeta_function()
-            (2/5*T^2 + 2/5*T + 1/5)/(2*T^2 - 3*T + 1)
+            (1/5*T^2 + 1/5*T + 1/10)/(T^2 - 3/2*T + 1/2)
         """
         P =  self.zeta_polynomial()
         q = (self.base_ring()).characteristic()

src/sage/combinat/sf/dual.py

@@ -735,7 +735,7 @@ def scalar_hl(self, x):
                 sage: h = m.dual_basis(scalar=zee)
                 sage: a = h([2,1])
                 sage: a.scalar_hl(a)
-                (t + 2)/(-t^4 + 2*t^3 - 2*t + 1)
+                (-t - 2)/(t^4 - 2*t^3 + 2*t - 1)
             """
             return self._dual.scalar_hl(x)
 

src/sage/combinat/sf/hall_littlewood.py

@@ -662,7 +662,7 @@ def scalar_hl(self, x, t = None):
                 sage: HLQ([2]).scalar_hl(HLQ([1,1]))
                 0
                 sage: HLP([2]).scalar_hl(HLP([2]))
-                1/(-t + 1)
+                -1/(t - 1)
             """
             parent = self.parent()
             if t is None:
@@ -837,13 +837,13 @@ def __init__(self, hall_littlewood):
             sage: HLQp = Sym.hall_littlewood().Qp()
             sage: s = Sym.schur(); p = Sym.power()
             sage: HLQ( HLP([2,1]) + HLP([3]) )
-            (1/(t^2-2*t+1))*HLQ[2, 1] + (1/(-t+1))*HLQ[3]
+            (1/(t^2-2*t+1))*HLQ[2, 1] - (1/(t-1))*HLQ[3]
             sage: HLQ(HLQp([2])) # indirect doctest
-            (t/(t^3-t^2-t+1))*HLQ[1, 1] + (1/(-t+1))*HLQ[2]
+            (t/(t^3-t^2-t+1))*HLQ[1, 1] - (1/(t-1))*HLQ[2]
             sage: HLQ(s([2]))
-            (t/(t^3-t^2-t+1))*HLQ[1, 1] + (1/(-t+1))*HLQ[2]
+            (t/(t^3-t^2-t+1))*HLQ[1, 1] - (1/(t-1))*HLQ[2]
             sage: HLQ(p([2]))
-            (1/(t^2-1))*HLQ[1, 1] + (1/(-t+1))*HLQ[2]
+            (1/(t^2-1))*HLQ[1, 1] - (1/(t-1))*HLQ[2]
         """
         HallLittlewood_generic.__init__(self, hall_littlewood)
 

src/sage/combinat/sf/jack.py

@@ -214,9 +214,9 @@ def P(self):
         ::
 
             sage: JP(JQ([2,1]))
-            ((t+2)/(2*t^3+t^2))*JackP[2, 1]
+            ((1/2*t+1)/(t^3+1/2*t^2))*JackP[2, 1]
             sage: JP(JQ([3]))
-            ((2*t^2+3*t+1)/(6*t^3))*JackP[3]
+            ((1/3*t^2+1/2*t+1/6)/t^3)*JackP[3]
             sage: JP(JQ([1,1,1]))
             (6/(t^3+3*t^2+2*t))*JackP[1, 1, 1]
 
@@ -266,13 +266,13 @@ def Q(self):
             sage: JQ = Sym.jack().Q()
             sage: JP = Sym.jack().P()
             sage: JQ(sum(JP(p) for p in Partitions(3)))
-            (1/6*t^3+1/2*t^2+1/3*t)*JackQ[1, 1, 1] + ((2*t^3+t^2)/(t+2))*JackQ[2, 1] + (6*t^3/(2*t^2+3*t+1))*JackQ[3]
+            (1/6*t^3+1/2*t^2+1/3*t)*JackQ[1, 1, 1] + ((2*t^3+t^2)/(t+2))*JackQ[2, 1] + (3*t^3/(t^2+3/2*t+1/2))*JackQ[3]
 
         ::
 
             sage: s = Sym.schur()
             sage: JQ(s([3])) # indirect doctest
-            (1/6*t^3-1/2*t^2+1/3*t)*JackQ[1, 1, 1] + ((2*t^3-2*t^2)/(t+2))*JackQ[2, 1] + (6*t^3/(2*t^2+3*t+1))*JackQ[3]
+            (1/6*t^3-1/2*t^2+1/3*t)*JackQ[1, 1, 1] + ((2*t^3-2*t^2)/(t+2))*JackQ[2, 1] + (3*t^3/(t^2+3/2*t+1/2))*JackQ[3]
             sage: JQ(s([2,1]))
             (1/3*t^3-1/3*t)*JackQ[1, 1, 1] + ((2*t^3+t^2)/(t+2))*JackQ[2, 1]
             sage: JQ(s([1,1,1]))
@@ -329,15 +329,15 @@ def J(self):
             sage: JJ = Sym.jack().J()
             sage: JP = Sym.jack().P()
             sage: JJ(sum(JP(p) for p in Partitions(3)))
-            1/6*JackJ[1, 1, 1] + (1/(t+2))*JackJ[2, 1] + (1/(2*t^2+3*t+1))*JackJ[3]
+            1/6*JackJ[1, 1, 1] + (1/(t+2))*JackJ[2, 1] + (1/2/(t^2+3/2*t+1/2))*JackJ[3]
 
         ::
 
             sage: s = Sym.schur()
             sage: JJ(s([3])) # indirect doctest
-            ((t^2-3*t+2)/(6*t^2+18*t+12))*JackJ[1, 1, 1] + ((2*t-2)/(2*t^2+5*t+2))*JackJ[2, 1] + (1/(2*t^2+3*t+1))*JackJ[3]
+            ((1/6*t^2-1/2*t+1/3)/(t^2+3*t+2))*JackJ[1, 1, 1] + ((t-1)/(t^2+5/2*t+1))*JackJ[2, 1] + (1/2/(t^2+3/2*t+1/2))*JackJ[3]
             sage: JJ(s([2,1]))
-            ((t-1)/(3*t+6))*JackJ[1, 1, 1] + (1/(t+2))*JackJ[2, 1]
+            ((1/3*t-1/3)/(t+2))*JackJ[1, 1, 1] + (1/(t+2))*JackJ[2, 1]
             sage: JJ(s([1,1,1]))
             1/6*JackJ[1, 1, 1]
         """
@@ -715,12 +715,12 @@ def _multiply(self, left, right):
             sage: JJ([1])^2              # indirect doctest
             (t/(t+1))*JackJ[1, 1] + (1/(t+1))*JackJ[2]
             sage: JJ([2])^2
-            (2*t^2/(2*t^2+3*t+1))*JackJ[2, 2] + (4*t/(3*t^2+4*t+1))*JackJ[3, 1] + ((t+1)/(6*t^2+5*t+1))*JackJ[4]
+            (t^2/(t^2+3/2*t+1/2))*JackJ[2, 2] + (4/3*t/(t^2+4/3*t+1/3))*JackJ[3, 1] + ((1/6*t+1/6)/(t^2+5/6*t+1/6))*JackJ[4]
             sage: JQ = SymmetricFunctions(FractionField(QQ['t'])).jack().Q()
             sage: JQ([1])^2              # indirect doctest
             JackQ[1, 1] + (2/(t+1))*JackQ[2]
             sage: JQ([2])^2
-            JackQ[2, 2] + (2/(t+1))*JackQ[3, 1] + ((6*t+6)/(6*t^2+5*t+1))*JackQ[4]
+            JackQ[2, 2] + (2/(t+1))*JackQ[3, 1] + ((t+1)/(t^2+5/6*t+1/6))*JackQ[4]
         """
         return self( self._P(left)*self._P(right) )
 
@@ -889,11 +889,12 @@ def _m_cache(self, n):
             sage: l(JP._m_to_self_cache[3])
             [([1, 1, 1], [([1, 1, 1], 1)]),
              ([2, 1], [([1, 1, 1], -6/(t + 2)), ([2, 1], 1)]),
-             ([3], [([1, 1, 1], 6/(t^2 + 3*t + 2)), ([2, 1], -3/(2*t + 1)), ([3], 1)])]
+             ([3], [([1, 1, 1], 6/(t^2 + 3*t + 2)), ([2, 1], -3/2/(t + 1/2)), ([3], 1)])]
             sage: l(JP._self_to_m_cache[3])
             [([1, 1, 1], [([1, 1, 1], 1)]),
              ([2, 1], [([1, 1, 1], 6/(t + 2)), ([2, 1], 1)]),
-             ([3], [([1, 1, 1], 6/(2*t^2 + 3*t + 1)), ([2, 1], 3/(2*t + 1)), ([3], 1)])]
+             ([3],
+              [([1, 1, 1], 3/(t^2 + 3/2*t + 1/2)), ([2, 1], 3/2/(t + 1/2)), ([3], 1)])]
         """
         if n in self._self_to_m_cache:
             return
@@ -1002,9 +1003,9 @@ def scalar_jack_basis(self, part1, part2 = None):
             sage: JP.scalar_jack_basis(Partition([2,1]), Partition([1,1,1]))
             0
             sage: JP._normalize_coefficients(JP.scalar_jack_basis(Partition([3,2,1]), Partition([3,2,1])))
-            (12*t^6 + 20*t^5 + 11*t^4 + 2*t^3)/(2*t^3 + 11*t^2 + 20*t + 12)
+            (6*t^6 + 10*t^5 + 11/2*t^4 + t^3)/(t^3 + 11/2*t^2 + 10*t + 6)
             sage: JJ(JP[3,2,1]).scalar_jack(JP[3,2,1])
-            (12*t^6 + 20*t^5 + 11*t^4 + 2*t^3)/(2*t^3 + 11*t^2 + 20*t + 12)
+            (6*t^6 + 10*t^5 + 11/2*t^4 + t^3)/(t^3 + 11/2*t^2 + 10*t + 6)
 
         With a single argument, takes `part2 = part1`::
 
@@ -1035,7 +1036,7 @@ def scalar_jack(self, x, t=None):
                 sage: JP = SymmetricFunctions(FractionField(QQ['t'])).jack().P()
                 sage: l = [JP(p) for p in Partitions(3)]
                 sage: matrix([[a.scalar_jack(b) for a in l] for b in l])
-                [  6*t^3/(2*t^2 + 3*t + 1)                         0                         0]
+                [3*t^3/(t^2 + 3/2*t + 1/2)                         0                         0]
                 [                        0     (2*t^3 + t^2)/(t + 2)                         0]
                 [                        0                         0 1/6*t^3 + 1/2*t^2 + 1/3*t]
             """
@@ -1195,9 +1196,14 @@ def _h_cache(self, n):
             [([1, 1], [([1, 1], 1), ([2], 2/(t + 1))]), ([2], [([2], 1)])]
             sage: JQp._h_cache(3)
             sage: l(JQp._h_to_self_cache[3])
-            [([1, 1, 1], [([1, 1, 1], 1), ([2, 1], 6/(t + 2)), ([3], 6/(2*t^2 + 3*t + 1))]), ([2, 1], [([2, 1], 1), ([3], 3/(2*t + 1))]), ([3], [([3], 1)])]
+            [([1, 1, 1],
+              [([1, 1, 1], 1), ([2, 1], 6/(t + 2)), ([3], 3/(t^2 + 3/2*t + 1/2))]),
+             ([2, 1], [([2, 1], 1), ([3], 3/2/(t + 1/2))]),
+             ([3], [([3], 1)])]
             sage: l(JQp._self_to_h_cache[3])
-            [([1, 1, 1], [([1, 1, 1], 1), ([2, 1], -6/(t + 2)), ([3], 6/(t^2 + 3*t + 2))]), ([2, 1], [([2, 1], 1), ([3], -3/(2*t + 1))]), ([3], [([3], 1)])]
+            [([1, 1, 1], [([1, 1, 1], 1), ([2, 1], -6/(t + 2)), ([3], 6/(t^2 + 3*t + 2))]),
+             ([2, 1], [([2, 1], 1), ([3], -3/2/(t + 1/2))]),
+             ([3], [([3], 1)])]
         """
         if n in self._self_to_h_cache:
             return
@@ -1291,7 +1297,7 @@ def coproduct_by_coercion( self, elt ):
             sage: Sym = SymmetricFunctions(QQ['t'].fraction_field())
             sage: JQp = Sym.jack().Qp()
             sage: JQp[2,2].coproduct()   #indirect doctest
-            JackQp[] # JackQp[2, 2] + (2*t/(t+1))*JackQp[1] # JackQp[2, 1] + JackQp[1, 1] # JackQp[1, 1] + ((4*t^3+8*t^2)/(2*t^3+5*t^2+4*t+1))*JackQp[2] # JackQp[2] + (2*t/(t+1))*JackQp[2, 1] # JackQp[1] + JackQp[2, 2] # JackQp[]
+            JackQp[] # JackQp[2, 2] + (2*t/(t+1))*JackQp[1] # JackQp[2, 1] + JackQp[1, 1] # JackQp[1, 1] + ((2*t^3+4*t^2)/(t^3+5/2*t^2+2*t+1/2))*JackQp[2] # JackQp[2] + (2*t/(t+1))*JackQp[2, 1] # JackQp[1] + JackQp[2, 2] # JackQp[]
         """
         h = elt.parent().realization_of().h()
         parent = elt.parent()

src/sage/combinat/sf/macdonald.py

@@ -782,12 +782,12 @@ def _s_to_self(self, x):
             sage: J = Sym.macdonald(t=2).J()
             sage: s = Sym.schur()
             sage: J._s_to_self(s[2,1])
-            ((-q+2)/(28*q-7))*McdJ[1, 1, 1] + (1/(-4*q+1))*McdJ[2, 1]
+            ((-1/28*q+1/14)/(q-1/4))*McdJ[1, 1, 1] - (1/4/(q-1/4))*McdJ[2, 1]
 
         This is for internal use only. Please use instead::
 
             sage: J(s[2,1])
-            ((-q+2)/(28*q-7))*McdJ[1, 1, 1] + (1/(-4*q+1))*McdJ[2, 1]
+            ((-1/28*q+1/14)/(q-1/4))*McdJ[1, 1, 1] - (1/4/(q-1/4))*McdJ[2, 1]
         """
         return self._from_cache(x, self._s_cache, self._s_to_self_cache, q = self.q, t = self.t)
 

src/sage/combinat/sf/sfa.py

@@ -4468,9 +4468,9 @@ def scalar_jack(self, x, t=None):
             [  0   0   0   0 384]
             sage: JQ = SymmetricFunctions(QQ['t'].fraction_field()).jack().Q()
             sage: matrix([[JQ(mu).scalar_jack(JQ(nu)) for nu in Partitions(3)] for mu in Partitions(3)])
-            [(2*t^2 + 3*t + 1)/(6*t^3)                         0                         0]
-            [                        0     (t + 2)/(2*t^3 + t^2)                         0]
-            [                        0                         0     6/(t^3 + 3*t^2 + 2*t)]
+            [(1/3*t^2 + 1/2*t + 1/6)/t^3                           0                           0]
+            [                          0 (1/2*t + 1)/(t^3 + 1/2*t^2)                           0]
+            [                          0                           0       6/(t^3 + 3*t^2 + 2*t)]
         """
         parent = self.parent()
         if t is None:

src/sage/dynamics/arithmetic_dynamics/affine_ds.py

@@ -701,9 +701,10 @@ def orbit(self, P, n):
             sage: A.<x,y> = AffineSpace(FractionField(R), 2)
             sage: f = DynamicalSystem_affine([(x-t*y^2)/x, t*x*y])
             sage: f.orbit(A(1, t), 3)
-            [(1, t), (-t^3 + 1, t^2), ((-t^5 - t^3 + 1)/(-t^3 + 1), -t^6 + t^3),
-            ((-t^16 + 3*t^13 - 3*t^10 + t^7 + t^5 + t^3 - 1)/(t^5 + t^3 - 1), -t^9 -
-            t^7 + t^4)]
+            [(1, t),
+             (-t^3 + 1, t^2),
+             ((t^5 + t^3 - 1)/(t^3 - 1), -t^6 + t^3),
+             ((-t^16 + 3*t^13 - 3*t^10 + t^7 + t^5 + t^3 - 1)/(t^5 + t^3 - 1), -t^9 - t^7 + t^4)]
         """
         Q = P
         if isinstance(n, list) or isinstance(n, tuple):

src/sage/dynamics/arithmetic_dynamics/endPN_automorphism_group.py

@@ -64,7 +64,7 @@ def automorphism_group_QQ_fixedpoints(rational_function, return_functions=False,
         sage: rational_function = (z^2 - 2*z - 2)/(-2*z^2 - 2*z + 1)
         sage: from sage.dynamics.arithmetic_dynamics.endPN_automorphism_group import automorphism_group_QQ_fixedpoints
         sage: automorphism_group_QQ_fixedpoints(rational_function, True)
-          [z, 2/(2*z), -z - 1, -2*z/(2*z + 2), (-z - 1)/z, -1/(z + 1)]
+          [z, 1/z, -z - 1, -z/(z + 1), (-z - 1)/z, -1/(z + 1)]
 
     ::
 
@@ -342,8 +342,8 @@ def PGL_repn(rational_function):
         sage: f = ((2*z-1)/(3-z))
         sage: from sage.dynamics.arithmetic_dynamics.endPN_automorphism_group import PGL_repn
         sage: PGL_repn(f)
-        [ 2 -1]
-        [-1  3]
+        [-2  1]
+        [ 1 -3]
     """
     if is_Matrix(rational_function):
         return rational_function
@@ -1083,7 +1083,7 @@ def three_stable_points(rational_function, invariant_list):
         sage: L = [[0,1],[4,1],[1,1],[1,0]]
         sage: from sage.dynamics.arithmetic_dynamics.endPN_automorphism_group import three_stable_points
         sage: three_stable_points(f,L)
-        [z, 4*z, 2/(2*z), 3/(2*z)]
+        [z, 4*z, 1/z, 4/z]
     """
     # define ground field and ambient function field
     if rational_function.parent().is_field():
@@ -1151,19 +1151,21 @@ def automorphism_group_FF_alg2(rational_function):
         sage: f = (3*z^3 - z^2)/(z-1)
         sage: from sage.dynamics.arithmetic_dynamics.endPN_automorphism_group import automorphism_group_FF_alg2
         sage: automorphism_group_FF_alg2(f)
-        [Univariate Polynomial Ring in w over Finite Field in b of size 7^2, [w, (3*b + 2)/((2*b + 6)*w)]]
+        [Univariate Polynomial Ring in w over Finite Field in b of size 7^2, [w, 5/w]]
 
     ::
 
         sage: R.<z> = PolynomialRing(GF(5^3,'t'))
         sage: f = (3456*z^(4))
         sage: from sage.dynamics.arithmetic_dynamics.endPN_automorphism_group import automorphism_group_FF_alg2
         sage: automorphism_group_FF_alg2(f)
-        [Univariate Polynomial Ring in w over Finite Field in b of size 5^6, [w,
-        (3*b^5 + 4*b^4 + 3*b^2 + 2*b + 1)*w, (2*b^5 + b^4 + 2*b^2 + 3*b + 3)*w,
-        (3*b^5 + 4*b^4 + 3*b^2 + 2*b)/((3*b^5 + 4*b^4 + 3*b^2 + 2*b)*w), (4*b^5
-        + 2*b^4 + 4*b^2 + b + 2)/((3*b^5 + 4*b^4 + 3*b^2 + 2*b)*w), (3*b^5 +
-        4*b^4 + 3*b^2 + 2*b + 3)/((3*b^5 + 4*b^4 + 3*b^2 + 2*b)*w)]]
+        [Univariate Polynomial Ring in w over Finite Field in b of size 5^6,
+         [w,
+          (3*b^5 + 4*b^4 + 3*b^2 + 2*b + 1)*w,
+          (2*b^5 + b^4 + 2*b^2 + 3*b + 3)*w,
+          1/w,
+          (3*b^5 + 4*b^4 + 3*b^2 + 2*b + 1)/w,
+          (2*b^5 + b^4 + 2*b^2 + 3*b + 3)/w]]
     """
     # define ground field and ambient function field
     if rational_function.parent().is_field():
@@ -1452,7 +1454,7 @@ def automorphisms_fixing_pair(rational_function, pair, quad):
         sage: L = [[4, 1], [2, 1]]
         sage: from sage.dynamics.arithmetic_dynamics.endPN_automorphism_group import automorphisms_fixing_pair
         sage: automorphisms_fixing_pair(f, L, False)
-        [(6*z + 6)/z, 4/(3*z + 3)]
+        [(6*z + 6)/z, 6/(z + 1)]
     """
     # define ground field and ambient function field
     if rational_function.parent().is_field():
@@ -1526,7 +1528,7 @@ def automorphism_group_FF_alg3(rational_function):
         sage: f = (3456*z^4)
         sage: from sage.dynamics.arithmetic_dynamics.endPN_automorphism_group import automorphism_group_FF_alg3
         sage: automorphism_group_FF_alg3(f)
-        [z, 3/(3*z)]
+        [z, 1/z]
     """
     # define ground field and ambient function field
     if rational_function.parent().is_field():

src/sage/dynamics/arithmetic_dynamics/projective_ds.py

@@ -2787,7 +2787,7 @@ def automorphism_group(self, **kwds):
             sage: R.<x,y> = ProjectiveSpace(QQ,1)
             sage: f = DynamicalSystem_projective([x^2-2*x*y-2*y^2, -2*x^2-2*x*y+y^2])
             sage: f.automorphism_group(return_functions=True)
-            [x, 2/(2*x), -x - 1, -2*x/(2*x + 2), (-x - 1)/x, -1/(x + 1)]
+            [x, 1/x, -x - 1, -x/(x + 1), (-x - 1)/x, -1/(x + 1)]
 
         ::
 
@@ -5736,12 +5736,31 @@ def automorphism_group(self, absolute=False, iso_type=False, return_functions=Fa
             sage: R.<x,y> = ProjectiveSpace(GF(3^2,'t'),1)
             sage: f = DynamicalSystem_projective([x^3,y^3])
             sage: f.automorphism_group(return_functions=True, iso_type=True) # long time
-            ([x, x/(x + 1), x/(2*x + 1), 2/(x + 2), (2*x + 1)/(2*x), (2*x + 2)/x,
-            1/(2*x + 2), x + 1, x + 2, x/(x + 2), 2*x/(x + 1), 2*x, 1/x, 2*x + 1,
-            2*x + 2, ((t + 2)*x + t + 2)/((2*t + 1)*x + t + 2), (t*x + 2*t)/(t*x +
-            t), 2/x, (x + 1)/(x + 2), (2*t*x + t)/(t*x), (2*t + 1)/((2*t + 1)*x +
-            2*t + 1), ((2*t + 1)*x + 2*t + 1)/((2*t + 1)*x), t/(t*x + 2*t), (2*x +
-            1)/(x + 1)], 'PGL(2,3)')
+            ([x,
+              x/(x + 1),
+              2*x/(x + 2),
+              2/(x + 2),
+              (x + 2)/x,
+              (2*x + 2)/x,
+              2/(x + 1),
+              x + 1,
+              x + 2,
+              x/(x + 2),
+              2*x/(x + 1),
+              2*x,
+              1/x,
+              2*x + 1,
+              2*x + 2,
+              (x + 2)/(x + 1),
+              2/x,
+              (2*x + 2)/(x + 2),
+              (x + 1)/(x + 2),
+              (2*x + 1)/x,
+              1/(x + 1),
+              1/(x + 2),
+              (2*x + 1)/(x + 1),
+              (x + 1)/x],
+             'PGL(2,3)')
 
         ::
 

src/sage/groups/perm_gps/permgroup.py

@@ -4132,10 +4132,10 @@ def molien_series(self):
 
             sage: G = SymmetricGroup(5)
             sage: G.molien_series()
-            1/(-x^15 + x^14 + x^13 - x^10 - x^9 - x^8 + x^7 + x^6 + x^5 - x^2 - x + 1)
+            -1/(x^15 - x^14 - x^13 + x^10 + x^9 + x^8 - x^7 - x^6 - x^5 + x^2 + x - 1)
             sage: G = SymmetricGroup(3)
             sage: G.molien_series()
-            1/(-x^6 + x^5 + x^4 - x^2 - x + 1)
+            -1/(x^6 - x^5 - x^4 + x^2 + x - 1)
 
         Some further tests (after :trac:`15817`)::
 
@@ -4212,7 +4212,7 @@ def poincare_series(self, p=2, n=10):
             (x^2 + 1)/(x^4 - x^3 - x + 1)
             sage: G = SymmetricGroup(3)
             sage: G.poincare_series(2,10)                              # optional - gap_packages
-            1/(-x + 1)
+            -1/(x - 1)
 
         AUTHORS:
 

src/sage/matrix/matrix2.pyx

@@ -331,7 +331,7 @@ cdef class Matrix(Matrix1):
             sage: v = vector([3,4*x - 2])
             sage: X = A \ v
             sage: X
-            ((-8*x^2 + 4*x + 9)/(10*x^3 + x), (19*x^2 - 2*x - 3)/(10*x^3 + x))
+            ((-4/5*x^2 + 2/5*x + 9/10)/(x^3 + 1/10*x), (19/10*x^2 - 1/5*x - 3/10)/(x^3 + 1/10*x))
             sage: A * X == v
             True